On All Real Zeros for a Class of Even Entire Functions

The present paper deals with a class of even entire functions of order $\rho =1$ and genus $\vartheta =0$ of the polynomials form, \[ \sum\limits_{m=0}^\infty {\frac{\left( {-1} \right)^m\Phi ^{\left( {2m} \right)}\left( 0 \right)}{\Gamma \left( {2m+1} \right)}x^{2m}} =\Phi \left( 0 \right)\prod\limits_{k=1}^\infty {\left( {1-\frac{x}{\ell _k }} \right)} , \] where $\Phi\left( 0 \right)\ne 0$, real numbers $x$, nonnegative integers $m$, and $\ell _k \ne 0$ are all of the nonzero roots with $\sum\limits_{k=1}^\infty {1/\left| {\ell _k } \right|} <\infty $ and natural numbers $k$. We provide an efficient criterion for the polynomials with only real zeros. We also prove that the conjecture of Jensen is our special case.

A Note on Few Interesting Approaches of Solving Equations to Find the Number of Real Zeros

Be it in the world of mathematics or real life, it is often rewarding to think out-of-the box while solving a problem. Accordingly, in this paper, our aim is to explore the various alternative approaches for solving algebraic equations and finding the number of real zeros. We will further delve deeper into the conceptual part of mathematics and understand how implementation of simple ideas can lead to an acceptable solution, which otherwise would have been tedious by considering the conventional approaches. In the pursuit of achieving the objective of this paper, we will consider few examples with full solutions coupled with precise explanation. It is also intended to leave something meaningful for the readers to explore further on their own. The fundamental objective of this paper is to emphasize on the importance of application of basic mathematical logic, concept of inequality, concept of domain and range of functions, concept of calculus and last but not the least the graphical approach in solving mathematical equations. As a further clarification on the scope of this paper, it is highly pertinent to bring to the understanding of the readers two important aspects - firstly, we will only deal with equations involving real variables; and secondly, this paper does not include topics related to number theory.

Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method

This work is concerned with the existence and uniqueness of boundary value problems defined on semi-infinite intervals. These kinds of problems seldom admit exactly known solutions and, therefore, the theoretical information on their well-posedness is essential before attempting to derive an approximate solution by analytical or numerical means. Our utmost contribution in this context is the definition of a numerical test for investigating the existence and uniqueness of solutions of boundary problems defined on semi-infinite intervals. The main result is given by a theorem relating the existence and uniqueness question to the number of real zeros of a function implicitly defined within the formulation of the iterative transformation method. As a consequence, we can investigate the existence and uniqueness of solutions by studying the behaviour of that function. Within such a context, the numerical test is illustrated by two examples where we find meaningful numerical results.

Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method

This work is concerned with the existence and uniqueness of boundary value problems defined on semi-infinite intervals. These kinds of problems seldom admit exactly known solutions and, therefore, the theoretical information on their well-posedness is essential before attempting to derive an approximate solution by analytical or numerical means. Our utmost contribution in this context is the definition of a numerical test for investigating the existence and uniqueness of solutions of boundary problems defined on semi-infinite intervals. The main result is given by a theorem relating the existence and uniqueness question to the number of real zeros of a function implicitly defined within the formulation of the iterative transformation method. As a consequence, we can investigate the existence and uniqueness of solutions by studying the behaviour of that function. Within such a context the numerical test is illustrated by two examples where we find meaningful numerical results.